We study essentially a Keller-Segel type chemotaxis system with nonlinear sensitivity (signifying by the exponent alpha) and production of signal (signifying by the exponent beta). We establish explicit relations between alpha, beta and the space dimension to ensure global- and local-in-time boundedness of classical solutions. In the attractive chemotaxis setting, our results cover the separate cases in the existing literature and they are critical by the quite known blow-up results. In the repulsive chemotaxis setting, we find that much wider regimes of alpha and beta compared to attraction case can ensure global existence and boundedness. Hence, our findings reveal strong “damping” effect of chemo-repulsion mechanism on boundedness, since blow-up has emerged if chemo-attraction mechanism is exerted instead. Furthermore, our 3-D local-in-time boundedness moves one step further towards the popular saying that there would no blow-up in the 3-D minimal negative chemotaxis model. The achievement of our goal is based on a new qualitative boundedness criterion and a new uniform-in-time compound space-time bound for the gradient of the chemical concentration. Besides, our results also relax the regularity requirement on the initial chemical concentration in the existing literature. This is based on a joint with K. Lin from Southwest University of Finance and Economics.
向田，2014年5月博士毕业于Tulane University，偏微分方程专业，2014年9月至2016年8月在中国人民大学数学科学研究院做博士后，2016年9月起为中国人民大学副教授，2018年起为硕士生导师。他的主要研究兴趣为非线性偏微分方程及其应用，非线性分析以及动力系统；已在SIAM JAM, JDE, Nonlinearity, EJAM, JNS, DCDS-A and B, CRM, Proc. AMS and NA-TMA/RWA, JMAA, CPA, JMP等杂志上发表二十余篇论文，为多个杂志审稿以及在第8届ICIAM以及第12届AIMS上各组织过一个研讨会。其研究得到中央高校科研启动基金，博士后基金一等以及国家自然科学基金青年基金的资助。